Existential characterizations of monadic NIP

TL;DR

This paper characterizes monadic NIP theories using existential formulas and establishes equivalences between NIP/stability of classes and their monadic versions, also linking non-monadic NIP to superexponential growth.

Contribution

It provides a new existential characterization of monadic NIP and connects NIP/stability properties of classes to their monadic counterparts, with growth rate implications.

Findings

Non-monadic NIP is witnessed by an existential configuration.

Hereditary classes are NIP/stable iff monadically NIP/stable.

Non-monadic NIP classes have superexponential growth.

Abstract

We show that if a universal theory is not monadically NIP, then this is witnessed by a canonical configuration defined by an existential formula. As a consequence, we show that a hereditary class of relational structures is NIP (resp. stable) if and only if it is monadically NIP (resp. monadically stable). As another consequence, we show that if such a class is not monadically NIP, then it has superexponential growth rate.